Whether you can download a private key depends entirely on how the certificate was requested and what type it is: Get an ACM exportable private key - AWS re:Post
Downloading Private Keys from AWS Certificate Manager (ACM) AWS Certificate Manager (ACM) is primarily designed to manage SSL/TLS certificates while keeping the private keys secure and non-exportable. However, recent updates have introduced ways to export private keys for specific certificate types. Understanding Export Eligibility aws certificate manager download private key